Solve 9x/x+1-9/x^2+x=50

$$\frac { 9 x } { x + 1 } - \frac { 9 } { x ^ { 2 } + x } = 50$$

Solve for x

$$x=-\frac{9}{41}\approx -0.219512195$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x+1,x^{2}+x.

$$x\times 9x-9=50x\left(x+1\right)$$

Multiply x and x to get x^{2}.

$$x^{2}\times 9-9=50x\left(x+1\right)$$

Use the distributive property to multiply 50x by x+1.

$$x^{2}\times 9-9=50x^{2}+50x$$

Subtract 50x^{2} from both sides.

$$x^{2}\times 9-9-50x^{2}=50x$$

Combine x^{2}\times 9 and -50x^{2} to get -41x^{2}.

$$-41x^{2}-9=50x$$

Subtract 50x from both sides.

$$-41x^{2}-9-50x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-41x^{2}-50x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -41x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=-50 ab=-41\left(-9\right)=369$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 369.

$$-1,-369 -3,-123 -9,-41$$

Calculate the sum for each pair.

$$-1-369=-370 -3-123=-126 -9-41=-50$$

The solution is the pair that gives sum -50.

$$a=-9 b=-41$$

Rewrite -41x^{2}-50x-9 as \left(-41x^{2}-9x\right)+\left(-41x-9\right).

$$\left(-41x^{2}-9x\right)+\left(-41x-9\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(41x+9\right)-\left(41x+9\right)$$

Factor out common term 41x+9 by using distributive property.

$$\left(41x+9\right)\left(-x-1\right)$$

To find equation solutions, solve 41x+9=0 and -x-1=0.

$$x=-\frac{9}{41} x=-1$$

Variable x cannot be equal to -1.

$$x=-\frac{9}{41}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x+1,x^{2}+x.

$$x\times 9x-9=50x\left(x+1\right)$$

Multiply x and x to get x^{2}.

$$x^{2}\times 9-9=50x\left(x+1\right)$$

Use the distributive property to multiply 50x by x+1.

$$x^{2}\times 9-9=50x^{2}+50x$$

Subtract 50x^{2} from both sides.

$$x^{2}\times 9-9-50x^{2}=50x$$

Combine x^{2}\times 9 and -50x^{2} to get -41x^{2}.

$$-41x^{2}-9=50x$$

Subtract 50x from both sides.

$$-41x^{2}-9-50x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-41x^{2}-50x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -41 for a, -50 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-41\right)\left(-9\right)}}{2\left(-41\right)}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-41\right)\left(-9\right)}}{2\left(-41\right)}$$

Multiply -4 times -41.

$$x=\frac{-\left(-50\right)±\sqrt{2500+164\left(-9\right)}}{2\left(-41\right)}$$

Multiply 164 times -9.

$$x=\frac{-\left(-50\right)±\sqrt{2500-1476}}{2\left(-41\right)}$$

Add 2500 to -1476.

$$x=\frac{-\left(-50\right)±\sqrt{1024}}{2\left(-41\right)}$$

Take the square root of 1024.

$$x=\frac{-\left(-50\right)±32}{2\left(-41\right)}$$

The opposite of -50 is 50.

$$x=\frac{50±32}{2\left(-41\right)}$$

Multiply 2 times -41.

$$x=\frac{50±32}{-82}$$

Now solve the equation x=\frac{50±32}{-82} when ± is plus. Add 50 to 32.

$$x=\frac{82}{-82}$$

Divide 82 by -82.

$$x=-1$$

Now solve the equation x=\frac{50±32}{-82} when ± is minus. Subtract 32 from 50.

$$x=\frac{18}{-82}$$

Reduce the fraction \frac{18}{-82} to lowest terms by extracting and canceling out 2.

$$x=-\frac{9}{41}$$

The equation is now solved.

$$x=-1 x=-\frac{9}{41}$$

Variable x cannot be equal to -1.

$$x=-\frac{9}{41}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x+1,x^{2}+x.

$$x\times 9x-9=50x\left(x+1\right)$$

Multiply x and x to get x^{2}.

$$x^{2}\times 9-9=50x\left(x+1\right)$$

Use the distributive property to multiply 50x by x+1.

$$x^{2}\times 9-9=50x^{2}+50x$$

Subtract 50x^{2} from both sides.

$$x^{2}\times 9-9-50x^{2}=50x$$

Combine x^{2}\times 9 and -50x^{2} to get -41x^{2}.

$$-41x^{2}-9=50x$$

Subtract 50x from both sides.

$$-41x^{2}-9-50x=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$-41x^{2}-50x=9$$

Divide both sides by -41.

$$\frac{-41x^{2}-50x}{-41}=\frac{9}{-41}$$

Dividing by -41 undoes the multiplication by -41.

$$x^{2}+\left(-\frac{50}{-41}\right)x=\frac{9}{-41}$$

Divide -50 by -41.

$$x^{2}+\frac{50}{41}x=\frac{9}{-41}$$

Divide 9 by -41.

$$x^{2}+\frac{50}{41}x=-\frac{9}{41}$$

Divide \frac{50}{41}, the coefficient of the x term, by 2 to get \frac{25}{41}. Then add the square of \frac{25}{41} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{50}{41}x+\left(\frac{25}{41}\right)^{2}=-\frac{9}{41}+\left(\frac{25}{41}\right)^{2}$$

Square \frac{25}{41} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{50}{41}x+\frac{625}{1681}=-\frac{9}{41}+\frac{625}{1681}$$

Add -\frac{9}{41} to \frac{625}{1681} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{50}{41}x+\frac{625}{1681}=\frac{256}{1681}$$

Factor x^{2}+\frac{50}{41}x+\frac{625}{1681}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{41}\right)^{2}=\frac{256}{1681}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{41}\right)^{2}}=\sqrt{\frac{256}{1681}}$$

Simplify.

$$x+\frac{25}{41}=\frac{16}{41} x+\frac{25}{41}=-\frac{16}{41}$$

Subtract \frac{25}{41} from both sides of the equation.

$$x=-\frac{9}{41} x=-1$$

Variable x cannot be equal to -1.

$$x=-\frac{9}{41}$$